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Omni-MATH / For a point P=(a,a²) in the coordinate plane, let l(P) denote the line passing through P with slope 2a. Consider the set of triangles with vertices of the form …

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problem

For a point P=(a,a2)P = (a,a^2) in the coordinate plane, let l(P)l(P) denote the line passing through PP with slope 2a2a. Consider the set of triangles with vertices of the form P1=(a1,a12),P2=(a2,a22),P3=(a3,a32)P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2), such that the intersection of the lines l(P1),l(P2),l(P3)l(P_1), l(P_2), l(P_3) form an equilateral triangle △\triangle. Find the locus of the center of △\triangle as P1P2P3P_1P_2P_3 ranges over all such triangles.
Plain-text mathematical notation (without MathML)
For a point P=(a,a²) in the coordinate plane, let l(P) denote the line passing through P with slope 2a. Consider the set of triangles with vertices of the form P₁=(a₁,a₁²),P₂=(a₂,a₂²),P₃=(a₃,a₃²), such that the intersection of the lines l(P₁),l(P₂),l(P₃) form an equilateral triangle △. Find the locus of the center of △ as P₁P₂P₃ ranges over all such triangles.
Original LaTeX notation
For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.

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Source and history

task_id
45065a54-f061-55b4-837b-2122e555b6a0
task_key
test--45065a54-f061-55b4-837b-2122e555b6a0
task_revision_id
1
config / split
/ test

Official source

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